Stokesphenomenon

Stokes phenomenon is the term used in asymptotic analysis to describe how the asymptotic expansion of a function solving a linear differential equation changes as one crosses certain rays in the complex plane, known as Stokes lines. In many problems the solution consists of several exponential terms, some of which dominate in a given sector, while others are subdominant. When a Stokes line is crossed, the coefficient of a subdominant exponential term changes, effectively turning on or off that term. In classical statements this change was considered abrupt; in refined treatments it is understood as a rapid but smooth transition.

The phenomenon arises prominently in the asymptotic solutions of differential equations with a large or small parameter, particularly in the WKB method near turning points or irregular singularities. If a solution is expressed as a combination of exponential factors, the relative dominance of these factors depends on the region of the complex plane. Stokes lines separate sectors where one exponential term dominates from those where another term is subdominant. The precise change is governed by Stokes multipliers, which can be determined by connection formulas or exponential asymptotics.

A classic illustration is the Airy equation, y'' - x y = 0, whose solutions Ai(x) and Bi(x) exhibit different exponential behavior in different sectors of the complex x-plane. Crossing a Stokes line causes a subdominant contribution to appear with a calculable multiplier, altering the asymptotic expansion accordingly.

Modern viewpoints emphasize smoothing of the transition (Berry smoothing), uniform and coupled asymptotics, and connections to resurgent analysis. Stokes phenomena are central to semiclassical approximations in physics and to the accurate asymptotics of special functions.